the harmonic balance method for bifurcation analysis of ...22ème congrès français de mécanique...

22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015

The Harmonic Balance Method for BifurcationAnalysis of Large-Scale Nonlinear Mechanical

Systems

T. DETROUX, L. RENSON, L. MASSET, G. KERSCHEN

Space Structures and Systems LaboratoryDepartment of Aerospace and Mechanical Engineering

University of Liège, Liège, BelgiumE-mail: tdetroux, l.renson, luc.masset, [email protected]

Abstract:The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutionsof nonlinear mechanical systems. The objective of this paper is to extend the method for bifurcationanalysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algo-rithm that combines the computation of the Floquet exponents with bordering techniques is developed. Anew procedure for the tracking of Neimark-Sacker bifurcations that exploits the properties of eigenvaluederivatives is also proposed. As an application, the frequency response of a structure spacecraft is stud-ied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonancecurves. This example illustrates how bifurcation tracking using the HB method can be employed as apromising design tool for detecting and eliminating such undesired behaviors.

Keywords: harmonic balance, continuation of periodic solutions, bifurcation detection and tracking,Floquet exponents, quasiperiodic oscillations, detached resonance curves.

1 IntroductionAs engineering structures are designed to be lighter and operate in more severe conditions, nonlinearphenomena such as amplitude jumps, modal interactions, limit cycle oscillations and quasiperiodic (QP)oscillations are expected to occur [1]. These are phenomena that can be frequently encountered in nor-mal operation regime, in space [2, 3] and aeroelastic [4] applications, or during GVT campaigns [5, 6].Classical excitation signals as swept-sine and harmonic tests can already reveal nonlinear behaviors. Formost of these nonlinear structures, bifurcations play a key role in the response dynamics; for example,fold bifurcations indicate a stability change for the periodic solutions, while QP oscillations are encoun-tered in the vicinity of Neimark-Sacker (NS) bifurcations. In that regard, it seems relevant to includeanalysis of bifurcations while performing a parametric study of the structure.

Different algorithms and numerical methods can be found in the literature for the computation of peri-odic solutions and their bifurcations. Most of them build on a continuation procedure [7], as a way tostudy the evolution of the solutions with respect to a certain parameter, e.g., the frequency of an external


excitation or a system parameter. Time domain methods, which deal with the resolution of a boundaryvalue problem (BVP), usually prove accurate for low-dimensional structures. When applied to largersystems however, their computational burden become substantial. For example, the shooting technique[8] requires numerous time integrations that can drastically slow down the speed of the algorithm. Meth-ods based on the orthogonal collocation, which uses a discretization of the BVP, are widely utilized insoftware for bifurcation detection and tracking like auto [9], colsys [10], content [11], matcont [12]or, more recently, coco [13]. In spite of its high accuracy and ability to address stiff problems, orthog-onal collocation is rarely employed to study large systems, which can be explained by the considerablememory space this method requires for the discretization of the problem.

Among all methods in the frequency domain, the harmonic balance (HB) method, also known as theFourier–Galerkin method, is certainly the most widely used. It approximates the periodic signals withtheir Fourier coefficients, which become the new unknowns of the problem. First implemented for ana-lyzing linear systems, the HB method was then successfully adapted to nonlinear problems, in electrical[14] and mechanical engineering [15, 16] for example. The main advantage of the HB method is thatit involves algebraic equations with less unknowns than the methods in the time domain, for problemsfor which low orders of approximation are sufficient to obtain an accurate solution; this is usually thecase if the regime of the system is not strongly nonlinear. For this reason, the HB method has receivedincreased attention for the last couple of years, which has led to numerous applications and adaptationsof the method [17, 18, 19] and to the development of a continuation package manlab [20, 21]. Thereaders can also refer to [22] for a comparison between the HB method and orthogonal collocation interms of convergence. Nevertheless, in spite of its performance and accuracy, to the authors’ knowledgethe HB method has never been extended to track bifurcations efficiently. In a previous work [23], theauthors laid down the foundations for this extension, through the application to a system with 2 degreesof freedom (DOFs). The purpose of this work is to provide implementation details to track bifurca-tions of larger structures, and to illustrate how the tracking procedure can reveal and explain unexpectednonlinear phenomena.

The paper is organized as follows. Section 2 recalls the theory of HB and its formulation in the frame-work of a continuation algorithm. In Section 3, Hill’s method is introduced for assessing the stabilityof periodic solutions and for detecting their bifurcations. The proposed bifurcation tracking procedureis then presented with its adaptations for fold, branch point and NS bifurcations. The overall method-ology is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinearvibration isolation device. Finally, the conclusions of the present study are summarized in Section 5.

2 Harmonic balance for periodic solutionsThis section first performs a brief review of the theory of the HB method. The method will be appliedto general non-autonomous nonlinear dynamical systems with n DOFs whose equations of motion are

Mx + Cx + Kx = fext(ω, t)− fnl(x, x) = f(x, x, ω, t) (1)

where M,C and K are the mass, damping and stiffness matrices respectively, x represents the displace-ments, the dots refer to the derivatives with respect to time t, fnl represents the nonlinear forces and fext


stands for the periodic external forces (harmonic excitation, for example) with frequency ω. The term fgathers both the external and nonlinear forces.

As recalled in the introduction, the periodic solutions x(t) and f(t) of equation (1) are approximated byFourier series truncated to the NH -th harmonic:

x(t) =cx0√2+

NH∑k=1

(sxk sin

(kωt

ν

)+ cxk cos

(kωt

ν

))(2)

f(t) =cf0√2+

NH∑k=1

(sfk sin

(kωt

ν

)+ cfk cos

(kωt

ν

))(3)

where sk and ck represent the vectors of the Fourier coefficients related to the sine and cosine terms,respectively. Here it is interesting to note that the Fourier coefficients of f (t), cfk and sfk , depend onthe Fourier coefficients of the displacements x (t), cxk and sxk . The integer parameter ν is introduced toaccount for some possible subharmonics of the external excitation frequency ω.

Substituting expressions (3) in equations (1) and balancing the harmonic terms with a Galerkin projec-tion yields the following nonlinear equations in the frequency domain

h(z, ω) ≡ A(ω)z − b(z) = 0 (4)

where A is the (2NH + 1)n × (2NH + 1)n matrix describing the linear dynamics of the system, z isthe vector containing all the Fourier coefficients of the displacements x(t), and b represents the vector ofthe Fourier coefficients of the external and nonlinear forces f(t). They have the following expressions:

A =

KK −

(ων

)2 M −ων C

ων C K −

(ων

)2 M. . .

K −(NH

ων

)2 M −NHων C

NHων C K −

(NH

ων

)2 M

(5)

z =

[(cx0)

T (sx1)T (cx1)

T . . .(

sxNH

)T (cxNH

)T]T

(6)

b =

[ (cf0)T (

sf1)T (

cf1)T

. . .(

sfNH

)T (cfNH

)T]T

(7)

In the time domain, expression (1) contains n equations while, in the frequency domain, expression (4)have (2NH + 1)n unknowns gathered in z. Expression (4) can be seen as the equations of amplitudeof (1): if z∗ is a root of (4), then the time signals x∗ constructed from z∗ with (3) are solutions of theequations of motion (1) and are periodic.


2.1 Analytical expression of the nonlinear terms and of the jaco-bian matrix of the system

Equation (4) is nonlinear and has to be solved iteratively (e.g., with a Newton-Raphson procedure, or withthe hybrid Powell nonlinear solver [24, 25]). At each iteration, an evaluation of b and of ∂h/∂z has to beprovided. When f can be accurately approximated with a few number of harmonics and when its analyt-ical sinusoidal expansion is known [26], or for some types of restoring force [27], analytical expressionsrelating the Fourier coefficients of the forces b and of the displacements z can be obtained together withthe expression of the Jacobian matrix of the system. Otherwise, the alternating frequency/time-domain(AFT) technique [28] can be used to compute b:

z FFT−1

−−−−−→ x(t) −→ f (x, x, ω, t) FFT−−−→ b(z) (8)

The Jacobian matrix of the system can then be computed through finite differences, which is computa-tionally demanding.

An efficient alternative consists in rewriting the inverse Fourier transform as a linear operatorΓ (ω). Firstproposed by Hwang [29], the method, also called trigonometric collocation, was adapted to the AFTtechnique [30], and has been widely used since then [31, 32, 33, 34]. Denoting N as the number of timesamples of a discretized period of oscillation, one defines vectors x and f containing the concatenatednN time samples of the displacements and the forces, respectively, for all DOFs:

x =[x1 (t1) . . . x1 (tN ) . . . xn (t1) . . . xn (tN )

]T(9)

f =[f1 (t1) . . . f1 (tN ) . . . fn (t1) . . . fn (tN )

]T(10)

The inverse Fourier transform can then be written as a linear operation:

x = Γ (ω) z (11)

with the nN × (2NH + 1)n sparse operator

Γ (ω) =

In ⊗

1/

√2

1/√2

...1/

√2

In ⊗

sin

(ωt1ν

)sin

(ωt2ν

)...

sin(ωtNν

)

In ⊗

cos

(ωt1ν

)cos

(ωt2ν

)...

cos(ωtNν

)

. . .

In ⊗

sin

(NH

ωt1ν

)sin

(NH

ωt2ν

)...

sin(NH

ωtNν

)

In ⊗

cos

(NH

ωt1ν

)cos

(NH

ωt2ν

)...

cos(NH

ωtNν

)

(12)

Figure 1 represents the inverse transformation matrix for the case n = 2, N = 64, and NH = 5.


2 4 6 8 10 12 14 16 18 20 22

20

40

60

80

100

120

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 1: Illustration of the inverse Fourier transformation matrix Γ (ω) for n = 2, N = 64, andNH = 5.

The direct Fourier transformation is written

z = (Γ (ω))+ x (13)

where + stands for the Moore-Penrose pseudoinverse Γ+ = ΓT(ΓΓT

)−1. The Fourier coefficients ofthe external and nonlinear forces are simply obtained by transforming the signals in the time domainback to the frequency domain:

b(z) = (Γ (ω))+ f (14)

The Jacobian matrix of expression (4) with respect to the Fourier coefficients z, denoted as hz , can becomputed as in [31, 32, 34]

hz =∂h∂z

= A − ∂b∂z

= A − ∂b∂ f

∂ f∂x

∂x∂z

= A − Γ+ ∂ f∂x

Γ (15)

In general, the derivatives of the forces with respect to the displacements in the time domain can beexpressed analytically, which leads to a very effective computation of the Jacobian matrix.

2.2 Continuation procedureUsually, the frequency response is to be computed in a range of frequencies, rather than for a singlefrequency ω. In this paper, a continuation procedure based on tangent predictions and Moore-Penrosecorrections, as in the software matcont [12], is coupled to HB. The search for a tangent vector t(i) atan iteration point

(z(i), ω(i)

)along the branch reads


[hz hω

tT(i−1)

]t(i) =

[01

](16)

where hω stands for the derivative of h with respect to ω,

hω =∂h∂ω

=∂A∂ω

z (17)

The last equation in (16) prevents the continuation procedure from turning back. For the first iterationof the procedure, the sum of the components of the tangent is imposed to be equal to 1.

The correction stage is based on Newton’s method. Introducing new optimization variables v(i,j) ini-tialized as v(i,1) = t(i), and y(i,j) =

[z(i,j) ω(i,j)

]T , the Newton iterations are constructed as follows:

y(i,j+1) = y(i,j) − G−1y

(y(i,j), v(i,j)

)G(

y(i,j), v(i,j))

v(i,j+1) = v(i,j) − G−1y

(y(i,j), v(i,j)

)R(

y(i,j), v(i,j)) (18)

with

G (y, v) =

[h (y)

0

], Gy (y, v) =

[hz (y) hω (y)

vT

],

R (y, v) =

[ [hz (y) hω (y)

]v

0

] (19)

3 Harmonic balance for bifurcation detection and tracking

3.1 Stability analysisThe continuation procedure presented in Section 2 does not indicate if a periodic solution is stable or not.In the case of time-domain methods, such as the shooting technique, a by-product of the continuationprocedure is the monodromy matrix of the system [8]. Its eigenvalues are termed the Floquet multipliersfrom which the stability of the solution can be deduced. For frequency-domain techniques, one can usea variant of the Floquet theory, the so-called Hill’s method, whose coefficients are also obtained as aby-product of the calculations. Hill’s method is known to give accurate results for a reasonable numberof harmonics NH , and to be effective for large systems [35].

In [16], von Groll et al. proposed the following quadratic eigenvalue problem proposed for finding theHill’s coefficients:

∆2λ2 +∆1λ+ hz = 0 (20)

When embedded in a continuation scheme, hz is already obtained from the correction stage (18-19).Since ∆1 and ∆2 are easily computed, the main computational effort amounts to solving a quadraticeigenvalue problem, which can be rewritten as a linear eigenvalue problem of double size

B1 − γB2 = 0 (21)


with

B1 =

[∆1 hz

−I 0

], B2 = −

[∆2 00 I

](22)

The coefficients λ are found among the eigenvalues of the (2NH + 1) 2n× (2NH + 1) 2n matrix

B = B−12 B1 (23)

=

[−∆−1

2 ∆1 −∆−12 hz

I 0

](24)

However, only 2n eigenvalues among the complete set λ approximate the Floquet exponents λ of thesolution x∗ [36]. The other eigenvalues are spurious and do not have any physical meaning; their numberalso increases with the number of harmonics NH . Moore [37] showed that the Floquet exponents λ arethe 2n eigenvalues with the smallest imaginary part in modulus. The diagonal matrix

B =

λ1

λ2

. . .λ2n

(25)

gathers the Floquet exponents and will play a key role for the detection and tracking of bifurcations inSections 3.2 and 3.3.

Eventually, the stability of a periodic solution can be assessed, i.e., if at least one of the Floquet exponentshas a positive real part, then the solution is unstable, otherwise it is asymptotically stable.

3.2 Detection of bifurcationsTo detect bifurcations, test functions ϕ are evaluated at each iteration of the numerical continuationprocess [12]. The roots of these test functions indicate the presence of bifurcations.

A fold bifurcation is simply detected when the iω-th component of the tangent prediction related to theactive parameter ω changes sign [38]. A suitable test function is thus

ϕF = tiω (26)

According to its algebraic definition [39], we note that a fold bifurcation is characterized by a rankdeficiency of 1 of the Jacobian matrix hz , with hω /∈ range (hz). Another (more computationally de-manding) test function is therefore

ϕF = |hz| (27)

Similarly, the Jacobian matrix is rank deficient for branch point (BP) bifurcations, with hω ∈ range (hz).They can be detected using the test function for fold bifurcations (27), but a more specific test functionis [38]


ϕBP =

∣∣∣∣∣ hz hω

tT

∣∣∣∣∣ (28)

The Neimark-Sacker (NS) bifurcation is the third bifurcation studied in this paper. It is detected when apair of Floquet exponents crosses the imaginary axis as a pair of complex conjugates. According to [40],the bialternate product of a m ×m matrix P, P⊙ = P ⊙ Im, is singular when P has a pair of complexconjugates crossing the imaginary axis. As a result, the test function for NS bifurcations is

ϕNS =∣∣∣B⊙

∣∣∣ (29)

To overcome the issue of computing determinants for large-scale systems, the so-called bordering tech-nique replaces the evaluation of the determinant of a matrix G with the evaluation of a scalar functiong which vanishes at the same time as the determinant [41]. The function g is obtained by solving thebordered system [

G pq∗ 0

][wg

]=

[01

](30)

where ∗ denotes conjugate transpose, and vectors p and q are chosen to ensure the nonsingularity of thesystem of equations. When G is almost singular, p and q are chosen close to the null vectors of G∗ andG, respectively. For instance, for NS bifurcations G = B⊙.

3.3 Bifurcation trackingOnce a bifurcation is detected, it can be tracked with respect to an additional parameter. To continuecodimension-1 bifurcations with respect to two parameters, such as, e.g., the frequency and amplitudeof the external forcing, the equation defining the bifurcation g = 0 is appended to (4)

h ≡ Az − b = 0

g = 0(31)

For fold and BP bifurcations, g is the solution of the bordered system (30) with G = hz . For NSbifurcations, G = B⊙ is considered in the bordered system.

During the continuation procedure, the computation of the derivatives of the additional equation is re-quired. As shown in [41], analytical expressions for the derivatives of g with respect to α, where α

denotes a component of z or one of the two active parameters, are found as

gα = −v∗Gαw (32)

where Gα is the derivative of G with respect to α, and where w and v comes from the resolution of thebordered system and its transposed version:

[G pq∗ 0

][wg

]=

[01

](33)


[G pq∗ 0

]∗ [ve

]=

[01

](34)

As a result, the only term that has to be evaluated is Gα. For fold and BP bifurcations,

Gα = hzα (35)

where hzα is the derivative of the Jacobian hz with respect to α and is computed through finite differ-ences.

For NS bifurcations,

Gα = ∂∂α

(B⊙

)=

(∂∂α

(B))

⊙=

∂λ1∂α

∂λ2∂α

. . .∂λ2n∂α

⊙

(36)

Finite differences could be used to compute the derivatives of the Floquet exponents. However, thismeans that the eigenvalue problem (21) has to be solved for each perturbation of the components of z,and for the perturbation of the two continuation parameters. This represents a total of n (2NH + 1)+ 2

resolutions of the eigenvalue problem per iteration, which is cumbersome for large systems. Instead, wepropose to compute the derivatives in (36) using the properties of eigenvalue derivatives demonstratedby Van der Aa et al. [42]. Denoting as Λ the eigenvector matrix of B, and ξ the localization vector con-taining the index of the 2n Floquet exponents λ among the eigenvalues λ, i.e., λi = λξi , the eigenvaluesderivatives can be computed as

∂λi

∂α=

(Λ−1∂B

∂αΛ

)(ξi,ξi)

(37)

An analytical expression relating the derivative of B with respect to α in function of hzα can be obtainedfrom Equation (24). As for fold and BP bifurcations, hzα is computed through finite differences.

4 Bifurcation analysis of a satellite structure

4.1 Description of the SmallSat spacecraftThe effectiveness of the proposed HB method is demonstrated using the SmallSat, a spacecraft structureconceived by EADS-Astrium (now Airbus Defence and Space). The spacecraft is 1.2m in height and1m in width. A prototype of the spacecraft with a dummy telescope is represented in Figure 2(a). Thenonlinear component, the so-called wheel elastomer mounting system (WEMS), is mounted on a bracketconnected to the main structure and loaded with an 8-kg dummy inertia wheel, as depicted in Figure 2(b).The WEMS device acts a mechanical filter which mitigates the on-orbit micro-vibrations of the inertiawheel through soft elastomer plots located between the metallic cross that supports the inertia wheeland the bracket. To avoid damage of the elastomer plots during launch, the axial and lateral motions of


Dummyinertiawheel

Dummytelescope

WEMSdevice

Main structure

(a)

X

Z

SmallSat

Inertia wheel

Bracket

Metallic

cross

Filtering

elastomer plot

Mechanical

stop

(b)

Figure 2: SmallSat spacecraft. (a) Photograph; (b) Schematic of the WEMS, the nonlinear vibrationisolation device.

the metallic cross are limited by four nonlinear connections labelled NC1 – 4. Each NC comprises twomechanical stops that are covered with a thin layer of elastomer to prevent metal-metal impacts.

A finite element model (FEM) was built to conduct numerical experiments. Linear shell elements wereused for the main structure and the instrument baseplate, and a point mass represented the dummytelescope. Proportional damping was considered for these components. As shown in Figure 3(a), themetallic cross of the WEMS was also modeled using linear shell elements whereas the inertia wheelwas seen as a point mass owing to its important rigidity. To achieve tractable calculations, the linear


elements of the FEM were condensed using the Craig-Bampton reduction technique. Specifically, theFEM was reduced to 10 internal modes and 9 nodes (excluding DOFs in rotation), namely both sides ofeach NC and the inertia wheel. In total, the reduced-order model thus contains 37 DOFs.

Each NC was then modeled using a trilinear spring in the axial direction (elastomer in traction/compressionplus two stops), a bilinear spring in the radial direction (elastomer in shear plus one stop) and a linearspring in the third direction (elastomer in shear). The values of the spring coefficients were identifiedfrom experimental data [2]. For instance, the stiffness curve identified for NC1 is displayed in Figure3(b). To avoid numerical issues, regularization with third-order polynomials was utilized in the closevicinity of the clearances to implement C1 continuity. The dissipation in the elastomer plots was mod-eled using lumped dashpots. We note that the predictions of the resulting nonlinear model were foundin good agreement with experimental data [2, 3].

For confidentiality, clearances and displacements are given through adimensionalized quantities through-out the paper.

4.2 Frequency response, Floquet exponents and bifurcation de-tection

The forced response of the satellite for harmonic forcing applied to the vertical DOF of the inertia wheelis computed using HB with NH = 9 harmonics and N = 1024 points per period. Figures 4 depictthe system’s frequency response curve at NC1-X and NC1-Z for two forcing amplitudes, F = 50 and155N. For a clear assessment of the effects of the nonlinearities, the response amplitudes are normal-ized with the forcing amplitude F in this figure. Because the normalized responses for the two forcingamplitudes coincide up to 23 Hz, one can conclude that the motion is purely linear in this frequencyrange. Conversely, the mode with a linear resonance frequency of 28.8Hz is greatly affected by theWEMS nonlinearities, which can be explained by the fact that this mode combines bracket deflectionwith WEMS motion [3].

Figure 5(a) presents a close-up of this nonlinear resonance at NC1-Z where stability and bifurcationsare also indicated. The evolution of the normalized harmonic coefficients

σi =ϕi∑NHk=0 ϕi

(i = 0, . . . , NH) (38)

withϕ0 =

cx0√2, ϕi =

√(sxi )

2 + (cxi )2 (i = 1, . . . , NH) (39)

is shown in Figure 5(b). From 20 to 23Hz, only the fundamental harmonic is present in the response.In the resonance region, the SmallSat nonlinearities activate other harmonics in the response. Evenharmonics contribute to the dynamics because of the asymmetric modeling of the NCs of the WEMS.From the figure, it is also clear that the 6th and higher harmonics have a negligible participation in theresponse; for this reason, NH = 5 is considered throughout the rest of the paper.


NC 4

NC 3

NC 2

NC 1

X

Y

Z

Inertia

wheel

Resto

ring f

orc

e a

t N

C1 (

N)

(a)

−2 −1 0 1 2−600

−400

−200

0

200

400

600

800

Relative displacement at NC1 [−]

Res

torin

g fo

rce

at N

C1

[N]

(b)

Figure 3: WEMS. (a) Modeling of the device using shell elements, a point mass, linear and nonlinearsprings. The linear and nonlinear springs are represented with squares and circles, respectively. (b)Identified stiffness curve of NC1 (in black) and fitting with a trilinear model (in red).


10 20 30 400

1

2

3

x 10−3

Frequency [Hz]

Nor

mal

ized

am

plitu

de x

/F [N

−1 ]

(a)

10 20 30 400

0.004

0.008

0.012

0.016

Frequency [Hz]

Nor

mal

ized

am

plitu

de x

/F [N

−1 ]

(b)

Figure 4: Normalized frequency response at NC1. (a) NC1-X; (b) NC1-Z. The solid and dashed linesrepresent forcing amplitude of F = 50N and F = 155N, respectively.


20 25 30 35 400.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Frequency [Hz]

Am

plitu

de [−

]

(a)

20 25 30 35 400

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Nor

mal

ized

har

mon

ic c

oeffi

cien

t σ i [−

]

i = 0i = 1i = 2i = 3i = 4i = 5i = 6i = 7i = 8i = 9

(b)

Figure 5: Frequency response at NC1-Z for F = 155N. (a) Displacement. Circle and triangle markersrepresent fold and NS bifurcations, respectively. The solid and dashed lines represent stable and unstablesolutions, respectively. (b) Harmonic coefficients.

The nonlinear resonance has a complex and rich topology. Not only the frequency response curve wasfound to fold backwards, but bifurcations were detected along the branch by the HB algorithm in Figure5(a). Fold bifurcations are generic for nonlinear resonances and are responsible for stability changes. NSbifurcations imply that further investigation of the dynamics should be carried out in the correspondingrange of frequencies, because a branch of quasiperiodic solutions bifurcates out from the main branch.To this end, the response to a swept-sine excitation with a forcing amplitudeF = 155N and with a sweeprate of 0.5Hz/min was computed with a Newmark time integration scheme. The sampling frequencywas chosen very high, i.e., 3000Hz, to guarantee the accuracy of the simulation in Figure 6(a). The


first observation is that overall the frequency response computed using HB provides a very accurateestimation of the envelope of the swept-sine response. In addition, a modulation of the displacement’senvelope is clearly noticed between the two NS bifurcations highlighted by HB, and the close-up inFigure 6(b) confirms the presence of quasiperiodic oscillations. Interestingly, the amplitudes associatedwith these oscillations are slightly larger than those at resonance, which shows the importance of a propercharacterization of these oscillations.

Figure 6(c) shows a subset of Hill’s coefficients λ and Floquet exponents λ obtained with Hill’s method(NH = 5 and N = 1024) for the stable periodic solution at 28Hz in Figure 6(a). Floquet exponents λTI

are also calculated from the monodromy matrix evaluated with a Newmark time integration scheme as in[8]; they serve as a reference solution. The comparison demonstrates that the actual Floquet exponentscorresponds to the Hill’s coefficients that are the closest ones to the real axis, which validates the sortingcriterion discussed in Section 3.1. The other coefficients are spurious, but they seem to be alignedaccording to the same pattern as that of the actual Floquet exponents. The location of all exponents inthe left-half plane indicates that the solution at 28Hz is indeed stable.

The evolution of Hill’s coefficients and Floquet exponents in the vicinity of the first NS bifurcationis given in Figures 7(a-b). Before the bifurcation, the Floquet exponents lie all in the left-half planein Figure 7(a), which indicates a stable solution. We stress that a pair of Hill’s coefficients has alreadycrossed the imaginary axis in this figure, which evidences that considering all Hill’s coefficients can thuslead to misjudgement. After the bifurcation, a pair of complex conjugates Floquet exponents now liein the right-half plane in Figure 7(b), which means that the system underwent a NS bifurcation and loststability. A similar scenario is depicted for the first fold bifurcation in Figures 7(c-d) with the differencethat a single Floquet exponent crosses the imaginary axis through zero.

4.3 Bifurcation trackingThe fold bifurcations revealed in the previous section are now tracked in thecodimension-2 forcing frequency-forcing amplitude space using the algorithm presented in Section 3.3.Figure 8(a) represents the resulting fold curve, together with the frequency responses of the system fordifferent forcing amplitudes. Very interestingly, the algorithm initially tracks the fold bifurcations of themain frequency response, but it then turns back to reveal a detached resonance curve (DRC), or isola.Such an attractor is rarely observed for real structures in the literature. Figure 8(b), which shows theprojection of the fold curve in the forcing amplitude-response amplitude plane, highlights that the DRCis created when F = 158N. The DRC then expands both in frequency and amplitude until F = 170Nfor which merging with the resonance peak occurs. Because the upper part of the DRC is stable, theresonance peak after the merging is characterized by a greater frequency and amplitude. This mergingprocess is further illustrated in Figure 9, where the responses to swept-sine excitations of amplitude 168,170, 172 and 174N are superimposed. Moving from F = 170N to F = 172N leads to a sudden ‘jump’in resonance frequency and amplitude.

Bifurcation tracking is not only useful for understanding the system’s dynamics and revealing additionalattractors, but it can also be used for engineering design. For instance, we use it herein to study theinfluence of the axial dashpot cax of the WEMS device on the quasiperiodic oscillations. Figure 10(a)depicts the NS curve in the codimension-2 forcing frequency-axial damping space to which frequencyresponses computed for cax = 63Ns/m (reference), 80Ns/m and 85Ns/m are superimposed. The pro-


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Figure 6: Bifurcations, quasiperiodic oscillations and Floquet exponents. (a) Comparison between HB(black) and a swept-sine response calculated using Newmark’s algorithm (blue) for F = 155N at NC1-Z. Circle and triangle markers depict fold and NS bifurcations, respectively. The solid and dashed linesrepresent stable and unstable solutions, respectively. (b) Close-up of the quasiperiodic oscillations. (c)Periodic solution at 28Hz: Hill’s coefficients λ (crosses), Floquet exponents λ obtained with Hill’smethod (circles), and Floquet exponents λTI obtained from the monodromy matrix (dots).

jection in the cax-response amplitude plane in Figure 10(b) shows that the two NS bifurcations, andhence quasiperiodic oscillations, can be completely eliminated for cax = 84Ns/m.

As a verification, Figure 11 shows the influence of cax on a displacement response for swept-sine exci-tations. At a forcing amplitude F = 155N and for an axial damping cax = 63Ns/m, the quasiperiodicoscillations represent the part of the response with the largest amplitude. Increasing cax up to 85Ns/m


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Figure 7: Floquet exponents in the vicinity of the fold and NS bifurcations. (a) Before the NS bifurcationat ω = 29.11Hz, stable region. (b) After the NS bifurcation at ω = 29.8Hz, unstable region. (c) Beforethe fold bifurcation at ω = 34.25Hz, stable region. (d) After the fold bifurcation at ω = 34.28Hz,unstable region. Hill’s coefficients λ and Floquet exponents λ obtained with Hill’s method are denotedwith crosses and circles, respectively.

eliminates the NS bifurcations which generate these disturbances, with a small impact on the frequencyof the resonance.


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Figure 10: Tracking of the NS bifurcations. (a) Three-dimensional space. Blue line: branch of NSbifurcations; black lines: frequency responses of the at NC1-Z for F = 155N and for cax = 63Ns/m,80Ns/m and 85Ns/m. Triangle markers depict NS bifurcations. (b) Two-dimensional projection of thebranch of NS bifurcations.


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Figure 11: SmallSat displacement response at NC1-Z node for swept-sine excitations of amplitudeF = 155N applied to the inertia wheel. The blue line and green line represent configuration withcax = 63Ns/m (reference) and 85Ns/m, respectively.


5 ConclusionsThe purpose of this paper was to extend harmonic balance from the computation of periodic solutionsto the tracking of bifurcations in codimension-2 parameter space. Particular attention was devoted tothe computational efficiency of the algorithm, which motivated the use of a bordering technique and thedevelopment of a new procedure for Neimark-Sacker bifurcations exploiting the properties of eigenvaluederivatives.

The method was demonstrated using a real satellite example possessing several mechanical stops. Specif-ically, we showed that bifurcation tracking of large-scale structures is now within reach and representsa particularly effective tool both for uncovering dynamical attractors and for engineering design in thepresence of nonlinearity.

AcknowledgmentsThe authors Thibaut Detroux, Luc Masset and Gaetan Kerschen would like to acknowledge the financialsupport of the European Union (ERC Starting Grant NoVib 307265). The author L. Renson is a Marie-Curie COFUND Postdoctoral Fellow of the University of Liège, co-funded by the European Union.

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